2 Answers
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Unless (like Asaf seems to be assuming) we're working at a much higher level of sophistication than it seems here, the elements of a set are simply numbers -- not expressions, representations or splotches of ink or chalk.

So when you write $\{10,(8+2)\}$ what you mean is that the number denoted by "$10$" is in the set, and the number denoted by "$(8+2)$" is in the set, and nothing else is. These two expression happen to denote the same number (which can also be described as "ten" or "one less than eleven" or 0x0A or "$1+1+1+1+1+1+1+1+1+1$"), so that number is the only element of the set in question, so its cardinality is $1$. Period.

It is somewhat common to understand sets intuitively as "lists of things where the order doesn't matter (and neither does repetitions)". This understanding can be misleading unless one is extremely careful about what a repetition is (not to speak of what a "list" means if it has infinitely many elements). What is really going on is:

A set of something that you can ask "is this one of your elements?" of for every "this" in the universe. The set consists of its yes/no answers to all of these questions, neither more nor less.

So when you write $\{10,(8+2)\}$ you're speaking of a set that answers "yes, $X$ is one of my elements" if and only if $X=10$ or $X=(8+2)$. But no matter what $X$ is, "$X=10$" and "$X=(8+2)$" are either both true or false, so that is the same as answering "yes, $X$ is one of my elements" if and only if $X=10$. And therefore $\{10,(8+2)\}$ is (a name for) the same set that $\{10\}$ is a name for. A set consists only of its answers, so when the answers are the same we're looking at the same set.